An equilateral triangle $ABC$ is cut from a thin solid sheet of wood. (see figure) $D, E$ and $F$ are the mid-points of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0.$ If the smaller triangle $DEF$ is removed from $ABC,$ the moment of inertia of the remaining figure about the same axis is $I.$ Then

An L-shaped object made of thin rods of uniform mass density is suspended with a string as shown in figure. If $AB=BC$ and the angle made by $AB$ with downward vertical is $\theta$, then,

A rod of length $50 \mathrm{cm}$ is pivoted at one end. It is raised such that if makes an angle of $30°$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $\mathrm{rad} {\mathrm{s}}^{-1}$) will be $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis (see figure). When released from the initial horizontal position, its instantaneous angular acceleration will be:

A circular disc $D_1$ of mass $M$ and radius $R$ has two identical discs $D_2$ and $D_3$ of the same mass $M$ and radius $R$ attached rigidly at its opposite ends (see figure.) The moment of inertia of the system about the axis $O{O}^{\text{'}}$, passing through the centre of $D_1$ as shown in the figure, will be:

A string is wound around a hollow cylinder of mass $5 \mathrm{kg}$ and radius $0.5 \mathrm{m}$. If the string is now pulled with a horizontal force of $40 \mathrm{N}$, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string):